Article Sidebar

Download
PDF
Statistic
Read Counter : 519 Download : 545

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Main Article Content

Abstract

For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.   

Keywords

Characterizations; Completely Monotonic Functions; Order Statistics; Record Values; Truncated Moment.

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following License

CC BY: This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.

 

Author Biographies

Mohammad Shakil, Miami Dade College, Hialeah Campus

Dr. M. Shakil, Ph.D., CStat (RSS, UK), PStat (ASA, USA), Member - AMS, ILAS & RGMIA
Professor of Mathematics, Department of Liberal Arts and Sciences - Mathematics
Miami Dade College, Hialeah Campus, 1780 West 49th Street, Suite 2325
Hialeah, Fl. 33012, USA
Telephone: (305) 237-8805 (Office); e-mail: mshakil@mdc.edu
Editor-in-Chief: POLYGON (MDC, Hialeah Campus, Florida, USA)
Member of the Editorial Board: JñÄnÄbha, JSTA, IJASP, PJSOR; Reviewer: Mathematical Reviews (AMS).
http://faculty.mdc.edu/mshakil/

Dr. Shakil is teaching as a Full-time Faculty in the Mathematics Dept. of Miami-Dade College, Hialeah Campus. Prior to joining MDC, Dr. Shakil taught at Florida International University, Nova Southeastern University, Johnson & Wales University, Broward Community College and several institutions in India and Nigeria. He holds a master's and a doctoral degree in mathematics from Patna University in India, and a master's degree in statistics from Florida International University. His research interests lie in the fields of numerical and functional analysis, statistics and probability, history of mathematics, mathematics and statistics education and curriculum development. He has several research publications to his credit in the field of mathematics, statistics and education.

Dr. Mohammad Ahsanullah, Rider University, Lawrenceville, New Jersey, USA

Professor Emeritus, Dept. of Management sciences, Rider University, Lawrenceville, New Jersey, USA

Dr. B. M. G. Kibria Kibria, Florida International University, Miami, Florida, USA

Professor, Dept. of Mathematical Sciences
Florida International University, Miami, Florida, USA

How to Cite
Shakil, M., Dr. Mohammad Ahsanullah, & Kibria, D. B. M. G. K. (2021). Characterizations of Some Probability Distributions with Completely Monotonic Density Functions . Pakistan Journal of Statistics and Operation Research, 17(1), 51-64. https://doi.org/10.18187/pjsor.v17i1.3491
Crossref
Scopus
Google Scholar
Europe PMC

References

  1. Abramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, USA.
  2. Ahsanullah, M. (1995). Record Statistics, Nova Science Publishers, New York, USA.
  3. Ahsanullah, M. (2017). Characterizations of Univariate Continuous Distributions, Atlantis Press, Paris, France.
  4. Ahsanullah, M., and Shakil, M. (2012). A note on the characterizations of Pareto distribution by upper record values, Communications of the Korean Mathematical Society, 27 (4), 835 - 842.
  5. Ahsanullah, M., Nevzorov,V. B., and Shakil, M. (2013). An Introduction to Order Statistics, Atlantis Press, Paris, France.
  6. Ahsanullah, M., Kibria, B. M. G., and Shakil, M. (2014). Normal and Student´s t Distributions and Their Applications, Atlantis Press, Paris, France.
  7. Ahsanullah, M., Shakil, M., and Kibria, B. G. (2015). Characterizations of folded student's t distribution, Journal of Statistical Distributions and Applications, 2(1), 15.
  8. Ahsanullah, M., and Shakil, M. (2015). Characterizations of continuous probability distributions occurring in physics and allied sciences by truncated moment, International Journal of Advanced Statistics and Probability, 3 (1), 100 - 114.
  9. Ahsanullah, M., Shakil, M., and Kibria, B. M. (2016). Characterizations of Continuous Distributions by Truncated Moment, Journal of Modern Applied Statistical Methods, 15 (1), 316 -331.
  10. Alzer, H., and Berg, C. (2002). Some classes of completely monotonic functions. Annales Academiae Scientiarum Fennicae Mathematica, 27 (2), 445 - 460.
  11. Alzer, H., and Berg, C. (2006). Some classes of completely monotonic functions, II, The Ramanujan Journal, 11 (2), 225 - 248.
  12. Arnold, B.C., Balakrishnan, and Nagaraja, H. N. (2005 ). First Course in Order Statistics, Wiley, New York, USA.
  13. Berg, C., and Forst, G. (1975). Potential theory on locally compact Abelian groups, Springer-Verlag, Berlin, Heidelberg, Germany.
  14. Chaudhry, M. A., and Zubair, S. M. (2002). On A Class of Incomplete Gamma Functions,
  15. with Applications, Chapman & Hall/CRC, Boca Raton, USA.
  16. David, H. A., and Nagaraja, H. N. (2003). Order Statistics, Third Edition, Wiley, New York, USA.
  17. Feller, W. (1971). An Introduction to Probability Theory and its Applications, vol. II, John Wiley and Sons, New York, USA..
  18. Galambos, J., and Kotz, S. (1978). Characterizations of probability distributions. A unified approach with an emphasis on exponential and related models, Lecture Notes in Mathematics, 675, Springer, Berlin, Germany.
  19. Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table of integrals, series, and products, Academic Press, Inc., San Diego, California, USA.
  20. Hausdorff, F. (1921). Summationsmethoden und momentfolgen. I, Mathematische Zeitschrift, 9 (1 - 2), 74 - 109.
  21. Ismail, M. E., Lorch, L., and Muldoon, M. E. (1986). Completely monotonic functions associated with the gamma function and its q-analogues, Journal of Mathematical Analysis and Applications, 116 (1), 1 - 9.
  22. Koudou, A. E., and Ley, C. (2014). Characterizations of GIG laws: A survey, Probability surveys, Volume 11, pp. 161 - 176. DOI: 10.1214/13-PS227.
  23. Kotz, S., and Shanbhag, D. N. (1980). Some new approaches to probability distributions, Advances in Applied Probability, 12, 903 - 921.
  24. McKay, A. T. (1932). A Bessel function distribution, Biometrika, 24 (1/2), 39 - 44.
  25. Miller, A. R. (1988). Incomplete Lipschitz-Hankel integrals of Macdonald functions, NRL Report 9112, Naval Research Laboratory, Washington D. C. 20375 - 5000, USA.
  26. Miller, K. S., and Samko, S. G. (2001). Completely monotonic functions, Integral Transforms and Special Functions, 12 (4), 389 - 402.
  27. Nagaraja, H. (2006). Characterizations of Probability Distributions, In Springer Handbook of Engineering Statistics (pp. 79 - 95), Springer, London, UK.
  28. Nevzorov, V. B. (2001). Records: Mathematical Theory, Translation of Mathematical Monograph, American Mathematical Society, Rhode Island, USA.
  29. Prudnikov, A. P., Brychkov, Y. A., and Marichev, O. I. (1986). Integrals and Series, Volume 2, Gordon and Breach Science Publishers, Amsterdam, Holland.
  30. Shakil, M., Ahsanullah, M., and Kibria, B. G. (2018). On the Characterizations of Chen's Two-Parameter Exponential Power Life-Testing Distribution, Journal of Statistical Theory and Applications, 17 (3), 393 - 407.
  31. Widder, D.V. (1941). The Laplace Transform, Princeton Univ. Press, Princeton, NJ, USA.
  32. Widder, D.V. (1971). An Introduction to Transform Theory, Academic Press, New York, USA.